Hydrodynamic instabilities and characteristic length. A key
observation in our simulations as well as other active fluid systems
[1]}, [2]}, [3]}, [4]} is that defect pairs are
generated along elongated cracks that themselves develop from regions
of high polar order. Here we performed a linear stability analysis for
a nematically ordered homogeneous base-state [5]}. As shown
in Fig. REF a, the analysis reveals a wave-number of
maximal growth, \(k_{cr}\) , along this direction, with \(k_{cr}\)  growing
with approximate linearity in \(\alpha =\alpha _{aa}+\alpha _{pa}\) . Also
the plane-wave vector of maximal growth is aligned with the nematic
director (\(\theta =0\)  in Fig. REF a inset). By contrast,
when solving a Stokes equation forced by a bulk stress, the
velocity-stress relation in Fourier space becomes
\({\hat{{\bf u}}}=\frac{i}{k}({\bf I}-{\hat{{\bf k}}}{\hat{{\bf k}}})({\hat{\mathbf {\Sigma }}}^e{\hat{{\bf k}}})\) .
Comparing to our surface model, the factor of \(k\) in the
denominator profoundly changes the nature of system stability, giving
that maximal growth occurs at \(k=0\)  for the bulk model, and so not
producing a characteristic length-scale
[6]}, [7]}. (This was also noted by [8]} in
their study of swimmers confined to immersed thin films, while
[9]} show that adding substrate friction changes
length-scale selection in 2D active nematic models.)
